Approximation Methods for Nonconvex Parabolic Optimal Control Problems Using Relaxed Controls

Abstract

We consider an optimal distributed control problem involving semilinear parabolic partial differential equations, with control and state constraints. Since no convexity assumptions are made, the problem is reformulated in relaxed form. The state equation is discretized using a finite element method in space and a θ -scheme in time, while the controls are approximated by blockwise constant relaxed controls. The first result is that, under appropriate assumptions, the properties of optimality, and of extremality and admissibility, carry over in the limit to the corresponding properties for the relaxed continuous problem. We also propose progressively refining discrete conditional gradient and gradient-penalty methods, which generate relaxed controls, for solving the continuous relaxed problem, thus reducing computations and memory. Numerical examples are given.